Journal of Marine Science and Technology, Vol. 13, No. 4, pp. 239-248 (2005)
239
A NOTE ON GRASHOF’S THEOREM
Wen-Tung Chang*, Chen-Chou Lin**, and Long-Iong Wu***
Key words: Grashof’s theorem, stationary configuration, uncertainty
configuration, planar four-bar linkage.
ABSTRACT
In this paper, Grashof’s theorem is justified from the perspective
of the occurrence of stationary configurations and uncertainty configurations of four-link chains. The link length relations at stationary
configurations and uncertainty configurations are examined in detail
based on the triangle inequality, and then the conditions for the
existence of a crank and a change point are deduced.
INTRODUCTION
The mobility problem has been of interest for a
long time in the area of mechanism design. Grashof [7]
first introduced a very simple rule to judge the rotatability
of links in four-bar linkages. Although the problem
drew the attention of many researchers [4, 8, 9], the rule,
known as Grashof’s theorem today, was not formally
proved until Paul [13] published his paper. In that
paper, Paul applied the triangle inequality to analyze the
mobility of four-link chains under their limit positions,
and proved that Grashof’s criterion is a necessary and
sufficient condition for the existence of a crank in a
four-link chain. He also proved the equality form of
Grashof’s criterion is a necessary and sufficient condition for the existence of a change point mechanism.
After the reassessment of Grashof’s criterion by Paul,
many works about the mobility of planar four-bar linkages were undertaken. Angeles and Callejas [1] derived
the constraining inequalities of displacement functions
of four-bar linkages, based on Freudenstein’s equation
[6]. The constraints, equivalent to Grashof’s criteria,
were applied to the optimal mobility design problems.
Midha et al. [12] utilized the triangle inequality concept
Paper Submitted 03/05/05, Accepted 06/23/05. Author for Correspondence:
Chen-Chou Lin. E-mail: [email protected].
*Ph.D. Candidate of Power Mechanical Engineering, National Tsing Hua
University, Hsinchu, Taiwan 30013, R.O.C.
**Professor of Mechanical and Mechatronic Engineering, National
Taiwan Ocean University, Keelung, Taiwan 20224, R.O.C.
***Professor of Power Mechanical Engineering, National Tsing Hua
University, Hsinchu, Taiwan 30013, R.O.C.
to formulate the mobility conditions for planar four-bar
linkages, and provided a graphical interpretation to
mobility determination. The concept can be extended to
a more complex planar linkage. Barker [3] classified
planar four-bar linkages into four classes of Grashof’s
category, four classes of non-Grashof’s category, and
six classes of change point category, based on the
properties of characteristic surfaces in positive octant
regions. Cohan and Yang [5] developed a transformation between Euclidean space and “parallel coordinates
system”, and analysed the mobility of linkages by the
ranges of the angular coordinates. Williams and
Reinholtz [18] proved that the polynomial discriminant
of four-bar linkages, after reductions, is equivalent to
Grashof’s law. Besides the mobility analysis of planar
four-bar linkages, some works were focused on the
areas of linkages with more than four bars and spatial
linkages. Ting [15] first introduced the Five-bar
Grashof’s criteria. The mobility criteria were further
extended to N-bar kinematic chains later by Ting [16]
and Ting and Liu [17]. In their derivations of rotatability
laws for N-bar linkages, Grashof’s theorem became a
very special case.
A mechanism may encounter a singular configuration under certain geometric conditions. At such
conditions, the instantaneous degree-of-freedom (d.o.
f.) or the transitory mobility of the mechanism will be
different from the result derived by the Grübler’s
equation. When a singular configuration occurs during
the motion, it represents that the relative motion state in
the mechanism is in a critical state; usually, the mobility
of some link (s) will become zero. Thus a study of the
geometric relations for linkages under singular conditions may help in understanding the mobility of the
linkages. Singular configurations can be further categorized into [19]: (a) stationary configuration, (b)
uncertainty configuration, and (c) dead-center configuration.
The main objective of this paper is to provide an
alternative proof to Grashof’s theorem from the perspective of the occurrence of singular configurations.
The geometric relations of four-link chains at stationary
configurations and uncertainty configurations are examined in detail, and then the conditions for the existence of a crank and a change point are deduced.
Journal of Marine Science and Technology, Vol. 13, No. 4 (2005)
240
GRASHOF’S THEOREM
ity (2). A classification of four-bar linkages is summarized in Table 1.
Grashof’s theorem [7] was first introduced in 1883
to deal with the mobility problem of planar four-bar
mechanisms, in particular, mechanisms with all revolute joints. Consider a four-link kinematic chain, let s
be the shortest link length, l be the longest link length,
p and q be the link lengths of the remaining two links,
where 1 > p ≥ q > s. Grashof stated that there exists at
least one link which can fully revolve with respect to the
other three links if
1+s≤p+q
(1)
and none of the four links can make a full revolution if
1+s>p+q
(2)
Inequality (1) is called the Grashof’s criterion or
Grashof’s inequality. Paul [13] further proved that the
Grashof’s criterion (1) is both necessary and sufficient
for the existence of at least one fully-rotatable link in a
four-bar linkage. A linkage is called a Grashof linkage
if it satisfies the Grashof’s criterion, while it is called a
non-Grashof linkage if the criterion is not satisfied. In
addition, a linkage is called a change point mechanism
or a Grashof neutral linkage if
1+s=p+q
(3)
At a change point, some instantaneous centers of
the mechanism can not be uniquely determined [19],
and the output behavior will become indeterminate.
Depending on the location of the shortest link, a
Grashof linkage can be categorized into three different
types:
(a) Crank-rocker mechanism: If the shortest link is adjacent to the ground, then the shortest link is fully
rotatable, and the other ground-adjacent link will
generate an oscillating motion.
(b) Double-crank mechanism: If the ground link is the
shortest, both ground-adjacent links are fully
rotatable.
(c) Double-rocker mechanism: If the coupler link is the
shortest, both ground-adjacent links will be in oscillating motion, and the coupler is fully rotatable.
Thus it is known that the shortest link is always
fully revolvable with respect to the other three links if
the Grashof’s criterion (1) is satisfied. The three movable links of a non-Grashof mechanism are always in
oscillating motion, hence a non-Grashof linkage is also
called a triple-rocker mechanism [13]. The non-Grashof
double-rocker mechanism named by Grashof [7] is in
fact a triple-rocker mechanism which satisfies inequal-
STATIONARY AND UNCERTAINTY
CONFIGURATIONS OF FOUR-LINK CHAINS
A stationary configuration belongs to one class of
the singular configurations of a kinematic chain, and so
does an uncertainty configuration. A study of these
configurations will provide a foundation to understand
the mobility conditions of linkage mechanisms.
1. Stationary configurations
A stationary configuration of a kinematic chain is
defined as the configuration when the joint of two
adjacent links becomes temporarily inactive. Under
such a condition, the two adjacent links are called the
stationary links, and the corresponding joint is called
the inactive joint [11, 19, 20]. At a stationary
configuration, there is no relative motion between the
two stationary links. One may regard the stationary
links together with the inactive joint as a sole link [2],
or a temporary rigid chain; and further, the instantaneous centers of a stationary link with respect to the
other links will be coincident. For example, in Figure 1
(a), I13 and I34 are coincident, and so are I 12 and I 24. A
mechanism will become locked if two adjacent stationary links are selected as the ground and the input link
respectively; however, the input link can revolve continuously if it never becomes stationary. Figure 1 shows
three different stationary configurations of a four-bar
linkage, where link 2 is the input link and link 4 is the
output link. Joint D in Figure 1(a), joint C in Figure 1
(b), and joint A in Figure 1(c) are the inactive joints
which lose their function at the corresponding
configurations. In Figure 1(a), the inactive joint D
coincides with the instant center I14, hence there is no
relative motion existing between links 1 and 4. It means
that the output link is stationary with respect to the
ground. In Figure 1(b), the inactive joint C coincides
with the instant center I 34 , hence there is no relative
motion existing between links 3 and 4. Nevertheless,
the input links in the above two cases are still movable.
Table 1. Classification of four-bar linkages [13, 14]
Case
l+s≤p+q
Shortest link
Type
1
2
3
4
5
<
<
<
=
>
Ground-adjacent
Ground
Coupler
Any
Any
Crank-rocker
Double-crank
Double-rocker
Change point
Triple-rocker
W.T. Chang et al.: A Note on Grashof’s Theorem
241
In Figure 1(c), the inactive joint A coincides with the
instant center I 12 , hence there is no relative motion
existing between links 1 and 2. It means that the input
link is stationary with respect to the ground and the
mechanism is locked. In summary, at a stationary
configuration of a four-link chain, if one of the stationary links is fixed temporarily, then the other stationary
link will be in its limit position and become motionless
at the moment. Since a link can not move in the same
direction after reaching its limit position, it indicates
the link can not make a full revolution. It is concluded
that two links in a four-link chain can not revolve
continuously with respect to each other if they are
adjacent to an inactive joint at a stationary
configuration. As shown in Figure 1, links 1 and 4 in
Figure 1(a), links 3 and 4 in Figure 1(b), and links 1 and
2 in Figure 1(c), all become relatively motionless at the
moment, and can not make a full revolution with respect
to each other.
Figure 2 shows all the possible stationary configurations of a four-link chain, where links k and j are
collinear, links h and i are the stationary links, and Ihi is
the inactive joint. At the stationary configurations, the
five instant centers (I hj, I hk, I ij, I ik , and I jk ) that do not
include the inactive joint (I hi) lie on the same line; and
further, the instant centers Ihk and Iik are coincident, and
so are Ihj and Iij. The stationary configuration shown in
Figure 2(a) is called the extended type, and the one
shown in Figure 2(b) is called the folded type. As
explained above, links h and i can not fully revolve with
respect to each other.
Fig. 1. Stationary configurations of four-bar linkages.
Fig. 2. All possible stationary configurations of a four-link chain.
2. Uncertainty configurations
In general, all links of a mechanism will undergo
a constraint motion after being assembled; however,
under some special circumstances, an uncertain state
may exist such that a link can possess multiple motion
paths. The uncertain state of a mechanism is called the
change point, and the configuration is called the uncertainty configuration [10, 19]. A mechanism possessing
such a property is called a change point mechanism.
The uncertainty configuration of a mechanism can be
identified by verifying the rank equality of the Jacobian
matrix and the extended Jacobian matrix of the motion
constraint equations [14, 19]. If the rank of the Jacobian
matrix is less than that of the extended Jacobian matrix,
it means the functional mapping between the primary
and secondary coordinates of the mechanism is not
unique. Physically, the angular velocity ratio between
certain two links will be zero versus zero, which represents an uncertain state. By using the concept of the
dead center configuration, Wu [19] introduced a systematic method of synthesizing the uncertainty configurations of mechanisms.
Figure 3(a) shows a four-bar linkage at an uncertainty configuration, whose links are collinear, and the
instant centers I13 and I24 are indeterminate. The length
of the two longer links is denoted by l, and that of the
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Journal of Marine Science and Technology, Vol. 13, No. 4 (2005)
two shorter links is denoted by s. If the linkage is a
parallelogram before reaching the uncertainty configuration as shown in Figure 3(b), the angular velocity ratio
between links 4 and 2 being +1, then the instant centers
I13 and I24 are located at the infinity on the line. On the
other hand, if the linkage is an anti-parallelogram before reaching the uncertainty configuration as shown in
Figure 3(c), the angular velocity ratio between links 4
and 2 being −1, then the instant centers I 13 and I 24 are
located between the two instant centers I 12 and I 14 ,
satisfying the relations I13I14 = (l − s)/2 and I12I24 = I14I24
, respectively. From Figure 3, it is shown that the instant
centers of two un-adjacent links can not be uniquely
determined when the four links are collinear. Figure 4
shows all the fully-collinear configurations of four-link
chains, where the instant centers Ihi, Ihk, Iij, and Ijk lie on
the same line, while the locations of I hj and I ik are
indeterminate. Among the figures, Figures 4(a) and 4
(b) are uncertainty configurations, while Figure 4(c) is
an immovable configuration.
LINK LENGTH RELATIONS AT STATIONARY
AND UNCERTAINTY CONFIGURATIONS
In the above section, it is shown that a four-link
chain will become a triangular enclosed area at a stationary configuration, while all four links will become
collinear at an uncertainty configuration. In this section
we shall discuss the link-length relations of the fourlink chains when the above configurations are
encountered.
Fig. 3. An uncertainty configuration of a four-bar linkage.
1. Stationary configurations
Let the lengths of the three sides of a triangle
formed by a four-link chain be u, v, and w. According
to the triangle inequality, the following relations can be
established.
u<v+w
(4)
v<u+w
(5)
w<u+v
(6)
Thus the link length relations of a four-link chain
at a stationary configuration must satisfy inequalities
(4), (5), and (6). When the four links with lengths l, p,
q, and s (1 > p ≥ q > s) are assembled into a triangle, the
following four conditions can exist:
(a) length relation l + s < p + q is valid.
(b) length relation l + s > p + q is valid.
(c) neither of above length relations is valid.
(d) length relation results in an unrealizable configuration.
The four conditions are illustrated as follows:
1.1. Length relation l + s < p + q is valid
Figure 5 shows two examples where the link length
Fig. 4. All possible fully-collinear configurations of a four-link chain.
W.T. Chang et al.: A Note on Grashof’s Theorem
relation 1 + s < p + q can be found in such triangles.
According to the triangle inequality, the link length of
the four-link chain shown in Figure 5(a) must satisfy the
relations:
(l + s) < p + q
(7)
p < (l + s) + q
(8)
q < (l + s) + p
(9)
Inequality (7) agrees with the link length relation
of Grashof’s criterion (Note that the equality form of the
criterion is discussed later). Inequalities (8) and (9) are
trivial since the length of any link must be less than the
sum of the other three link lengths. In the same manner,
the link-length of the four-link chain shown in Figure 5
(b) must satisfy the relations:
243
(p + q) < l + s
(13)
s < l + (p + q)
(14)
l < s + (p + q)
(15)
Inequality (13) satisfies the relation l + s > p + q.
Inequalities (14) and (15) are trivial relations. Similarly,
the link lengths of the four-link chain shown in Figure
6(b) must satisfy the relations:
(l − q) < p + s
(16)
s < (l − q) + p
(17)
p < (l − q) + s
(18)
(q − s) < l + p
(10)
Again, inequalities (16) and (17) are also trivial
relations. Inequality (18) satisfies the relation l + s > p
+ q.
p < l + (q − s)
(11)
1.3. Neither of the above length relations is valid
l < p + (q − s)
(12)
Figure 7 shows an example where neither the
length relation l + s < p + q nor l + s > p + q can be found
in such a triangle. According to the triangle inequality,
the link lengths of the four-link chain shown in Figure
Again, inequalities (10) and (11) are trivial
relations. Inequality (12) agrees with the link length
relation of Grashof’s criterion.
1.2. Length relation l + s > p + q is valid
Figure 6 shows two examples where the link length
relation l + s > p + q can be found in such triangles.
According to the triangle inequality, the link-length of
the four-link chain shown in Figure 6(a) must satisfy the
relations:
Fig. 6. Examples of stationary configurations whose length relations
satisfy relation l + s < p + q.
Fig. 5. Examples of stationary configurations whose length relations
satisfy l + s > p + q.
Fig. 7. An example of stationary configuration whose length relation
does not satisfy l + s < p + q or l + s > p + q.
Journal of Marine Science and Technology, Vol. 13, No. 4 (2005)
244
7 must satisfy the relations:
2.1. Length relation l + s = p + q is valid
(p + s) < l + q
(19)
q < l + (p + s)
(20)
l < q + (p + s)
(21)
Figure 9 shows two examples of the uncertainty
configurations satisfying the length relation l + s = p +
q. In Figure 9(a), the distance between the two endpoints A and B is
AB = l + s = p + q
Since p < l and s < q, inequality (19) is trivial, and
so are inequalities (20) and (21). In summary, it is
shown that neither the relation l + s < p + q nor l + s >
p + q exist in such a triangle.
1.4. Length relation results in an unrealizable configuration
Figure 8 shows two examples where the link length
relations will result in unrealizable triangles. The fourlink chain shown in Figure 8(a) must satisfy the relation
l + q < p + s. However, the configuration with the length
relation can not exist since l > p and q > s. The
configuration shown in Figure 8(b) is also not realizable
since l > s. Hence the four-link chain with the above
length relations can not form such triangles.
2. Uncertainty configurations
At an uncertainty configuration the four links in a
four-link chain will become collinear. As the four links
with lengths l, p, q, and s (l > p ≥ q > s) are assembled
into a closure with zero area, the following conditions
may occur:
(a) length relation l + s = p + q is valid.
(b) length relation results in an unrealizable zero area
closure.
The two conditions are illustrated as follows:
Fig. 8. Examples of stationary configurations whose length relations
result in unrealizable configurations.
(22)
Eq. (22) satisfies the length relation l + p = q + s.
Similarly, in Figure 9(b), the distance between two endpoints A and B is
AB = l = p + q − s
(23)
Thus Eq. (23) also satisfies the length relation l +
s = p + q.
2.2. Length relation results in an unrealizable zero area
closure
Figure 10 shows three examples that the length
relations will result in unrealizable zero area closures.
The configuration shown in Figure 10(a) must satisfy
the relation l + p = q + s. However, the configuration
with the length relation can not exist since l > q and p <
s. The configurations shown in Figures 10(b) and (c)
must satisfy the relations l + q = p + s and l + s = p + q,
respectively. However, the two configurations also can
not exist since l is not the shortest link length. Thus the
above configurations are physically unrealizable. Three
types of link length relations resulting in unrealizable
uncertainty configurations are summarized as follows:
(a) l + p = q + s is satisfied, while l > p ≥ q > s is violated.
Fig. 9. Examples of the uncertainty configurations whose length
relations satisfy l + s = p + q.
W.T. Chang et al.: A Note on Grashof’s Theorem
(b) l + q = p + s is satisfied, while l > p ≥ q > s is violated.
(c) l + s = p + q is satisfied, while l > p ≥ q > s is violated.
Note that the above length relations may still be
valid if the constraints l > p ≥ q > s are modified. This
issue is to be discussed later.
245
In this section, Grashof’s criterion is proved by
first listing all the stationary configurations occurring
in a four-link chain, and then establishing the valid
length relations based on the triangle inequality. Tables
2, 3, and 4 list the link length arrangements of Types I,
II, and III four-link chains at their stationary
configurations, respectively. Each table contains two
groups: (a) the shortest link belongs to one of the
collinear links, and (b) the shortest link belongs to one
of the stationary links. In cases 3, 4, 7, and 8 of Table
2, the shortest link belongs to one of the collinear links.
The length relation can be derived from case 8 using the
triangle inequality, as illustrated in subsection 1.1 of
above section. The length relation is trivial in case 3
since s < q and p < l are always true. Similarly, trivial
length relations can be found in case 4. The length
relation is unrealizable in case 7 since l < q + (s − p) is
always false. In cases 1, 2, 5, and 6 of Table 2, the
shortest link belongs to one of the stationary links. The
length relation l + s > p + q can be derived from case 5
using the triangle inequality, as illustrated in subsection
1.2 of above section. The unrealizable length relations
can be derived in cases 1, 2, and 6 when one follows the
same logic as that in case 7. As a summary from Table
2, it can be shown that when the length relation l + s <
p + q is true (case 8), the shortest link must belong to one
of the collinear links at a stationary configuration; and
the shortest link can make a full revolution with respect
to the adjacent collinear link since it will never become
a stationary link during the complete cycle of the motion.
On the other hand, when the length relation l + s > p +
Fig. 10. Examples of uncertainty configurations whose length relations
result in unrealizable configurations.
Fig. 11. All possible link permutations of a four-link chain.
PROOF OF GRASHOF’S THEOREM
Figure 11 shows all the link permutations of a
four-link chain having link-lengths l, p, q, and s. Among
them, the kinematic chains in Figures 11(a) and 11(b),
11(c) and 11(d), 11(e) and 11(f) are topologically
identical. Thus only the chains in Figures 11(a), 11(c),
and 11(e) will be considered. In this paper, the four-link
chains in Figures 11(a), 11(c), and 11(e) are called Type
I (ordered by s − p − l − q), Type II (ordered by s − l −
p − q), and Type III (ordered by s − l − q − p),
respectively.
1. Proof of Grashof’s criterion
246
Journal of Marine Science and Technology, Vol. 13, No. 4 (2005)
q is true (case 5), the shortest link must belong to one of
the stationary links at a stationary configuration; and
the shortest link can not make a full revolution.
Accordingly, the same conclusions can be derived from
Table 3 and Table 4 for Type II and Type III four-link
chains, respectively. Finally, when the length relation
l + s = p + q is satisfied, the shortest link is collinear with
the other three links at a singular configuration; if the
shortest link is an input link, it can revolve continuously
with respect to the adjacent link without any problem,
Table 2. Stationary configurations of Type I four-link chains
but if the shortest link is not an input link, its motion
path will be indeterminate at the collinear configuration.
Nevertheless, the relative motion between the shortest
link and its adjacent link will be a full revolution.
From the above deductions, two cases can be established as follows:
Case 1: If and only if the length relation l + s ≤ p + q is
true, the shortest link must be one of the collinear links at a stationary configuration, and
can fully revolve with respect to the adjacent
collinear link.
Case 2: If and only if the length relation l + s > p + q is
true, the shortest link must be one of the stationary links at a stationary configuration, and can
not fully revolve with respect to the adjacent
Table 4. Stationary configurations of Type III four-link chains
Table 3. Stationary configurations of Type II four-link chains
Table 5. Uncertainty configurations of Type I four-link chains
W.T. Chang et al.: A Note on Grashof’s Theorem
247
Table 6. Uncertainty configurations of Type II four-link chains
Table 7. Uncertainty configurations of Type III four-link chains
stationary link.
A lemma proved in Paul’s paper [13] stated that: a
link in a four bar chain can revolve relative to another
link if, and only if, some link revolves relative to an
adjacent link. By adding the lemma to the above
deductions, the two cases can be rephrased as follows:
If and only if the length relation satisfies l + s ≤ p
+ q in a four-link chain, the shortest link can always
fully revolve with respect to the other links; If and only
if the length relation satisfies l + s > p + q, none of the
links can revolve continuously with respect to the other
links.
Thus the proof of Grashof’s criterion is completed.
mechanism when a ground link is chosen. The above
conclusion can be rephrased as follows:
A four-link chain will become a change point
mechanism after the process of the kinematic inversion
if, and only if, the length relation satisfies l + s = p + q.
There are some special cases that the change point
conditions occur in a four-link chain while the constraints l > p ≥ q > s are violated. Cases 3, 5, and 6 in
Table 5 satisfy l + p = q + s while violating l > p ≥ q >
s; Case 2 in Tables 5 and 6, and cases 3 through 6 in
Table 7 satisfy l + p = q + s while violating l > p ≥ q >
s. The only reasonable solution to the above cases is
that the link lengths satisfy l = p = q = s. A mechanism
satisfied such condition is called a triple change point
mechanism by Barker [3].
Furthermore, case 1 in Table 5, cases 3 through 6
in Table 6, and case 2 in Table 7 satisfy l + q = p + s while
violating l > p ≥ q > s. The above cases will be valid if
l = p ≥ q = s or l = p = q = s is satisfied. When the lengths
satisfy l = p ≥ q = s, if the order of the link lengths is l
− s − l − s, then the linkage is a parallelogram or an antiparallelogram (a bowtie); if the order of the link lengths
is l − l − s − s, then the linkage is a kite type. When the
lengths satisfy l = p = q = s, the linkage is a triple change
point linkage.
2. Proof of change point condition
It is shown that at an uncertainty configuration of
a four-link chain, the four links will become collinear
and the instant centers of any two un-adjacent links can
not be uniquely determined. Such a four-link chain will
become a change point mechanism when a ground link
is chosen. Tables 5, 6, and 7 list the link length
arrangements of Types I, II, and III four-link chains at
their uncertainty configurations, respectively. In Table
5, the length relation l + s = p + q can be derived in case
4, while the length relations are unrealizable in the
remaining five cases. In Tables 6 and 7, the length
relation l + s = p + q can be derived in case 1, while the
length relations are unrealizable in the other ten cases.
Summarized from Tables 5, 6, and 7, a conclusion can
be made as follows:
The uncertainty configuration of a four-link chain
will occur if, and only if, the length relation satisfies l
+ s = p + q.
As mentioned above, a four-link chain possessing
an uncertain configuration will become a change point
CONCLUSION
In this paper, Grashof’s theorem or the mobility
condition of planar four-bar linkages is justified from
the perspective of the occurrence of singular configurations of four-link chains. It is shown that a four-link
chain will become a triangular enclosed area at a stationary configuration, while all four links will become
collinear at an uncertainty configuration. By permuting
the link lengths in the stationary configurations of four-
248
Journal of Marine Science and Technology, Vol. 13, No. 4 (2005)
link chains, all cases of stationary configurations are
generated and categorized in Tables 2, 3, and 4.
Likewise, all cases of uncertainty configurations are
generated and categorized in Tables 5, 6, and 7. An
examination of the triangle inequality on all cases of
Tables 2, 3, and 4 has shown that, after excluding the
trivial cases and unrealizable cases, the valid cases will
lead to the same conclusions as Grashof’s theorem. An
examination of the length equality between two end
points on the cases of Tables 5, 6, and 7 has shown that,
after excluding the trivial cases and unrealizable cases,
the valid cases will lead to the change point condition.
6.
7.
8.
9.
10.
11.
NOMENCLATURE
A, B, C, D
I mn
l
ln
p
q
s
u, v, w
revolute joints
instant center of links labeled m and n
longest link length of a four-link chain
length of a link labeled n
second longest link length of a four-link
chain
second shortest link length of a four-link
chain
shortest link length of a four-link chain
side lengths of a triangle
12.
13.
14.
15.
REFERENCES
16.
1. Angeles, J. and Callejas, M., “An Algebraic Formulation
of Grashof’s Mobility Criteria with Application to Linkage Optimization Using Gradient-Dependent Methods,”
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